If the vectors `3 vec p + vec q`; `5 vec p - 3 vecq` and `2 vec p + vec q`; `4 vec p - 2 vec q` are pairs of mutually perpendicular then sin( `vec p`,`vec q)` is :

Options

A. a) `(sqrt(55))/(4)`
B. b) `(sqrt(55))/(8)`
C. c) `(3)/(16)`
D. d) `(sqrt(247))/(16)`

Answer

To solve the problem, we need to find the sine of the angle between two vectors \(\vec{p}\) and \(\vec{q}\) given that certain pairs of vectors formed from these are mutually perpendicular.

### Step-by-Step Solution:

1. **Identify the vectors**: We have two pairs of vectors:
- Pair 1: \( \vec{A} = 3\vec{p} + \vec{q} \) and \( \vec{B} = 5\vec{p} - 3\vec{q} \)
- Pair 2: \( \vec{C} = 2\vec{p} + \vec{q} \) and \( \vec{D} = 4\vec{p} - 2\vec{q} \)

2. **Set up the first dot product**: Since \(\vec{C}\) and \(\vec{D}\) are perpendicular, we have:
\[ \vec{C} \cdot \vec{D} = 0 \]
This gives:
\[ (2\vec{p} + \vec{q}) \cdot (4\vec{p} - 2\vec{q}) = 0 \]

3. **Calculate the dot product**:
\[ 2\vec{p} \cdot 4\vec{p} + 2\vec{p} \cdot (-2\vec{q}) + \vec{q} \cdot 4\vec{p} + \vec{q} \cdot (-2\vec{q}) = 0 \]
This simplifies to:
\[ 8|\vec{p}|^2 - 4\vec{p} \cdot \vec{q} + 4\vec{p} \cdot \vec{q} - 2|\vec{q}|^2 = 0 \]
Thus:
\[ 8|\vec{p}|^2 - 2|\vec{q}|^2 = 0 \]

4. **Rearranging the equation**:
\[ 8|\vec{p}|^2 = 2|\vec{q}|^2 \implies 4|\vec{p}|^2 = |\vec{q}|^2 \]
This implies:
\[ |\vec{q}| = 2|\vec{p}| \]
(Mark this as Equation 1)

5. **Set up the second dot product**: Now consider the second pair:
\[ \vec{A} \cdot \vec{B} = 0 \]
This gives:
\[ (3\vec{p} + \vec{q}) \cdot (5\vec{p} - 3\vec{q}) = 0 \]

6. **Calculate the dot product**:
\[ 3\vec{p} \cdot 5\vec{p} + 3\vec{p} \cdot (-3\vec{q}) + \vec{q} \cdot 5\vec{p} + \vec{q} \cdot (-3\vec{q}) = 0 \]
This simplifies to:
\[ 15|\vec{p}|^2 - 9\vec{p} \cdot \vec{q} + 5\vec{p} \cdot \vec{q} - 3|\vec{q}|^2 = 0 \]
Thus:
\[ 15|\vec{p}|^2 - 4\vec{p} \cdot \vec{q} - 3|\vec{q}|^2 = 0 \]

7. **Substituting Equation 1 into the second equation**:
Substitute \( |\vec{q}|^2 = 4|\vec{p}|^2 \):
\[ 15|\vec{p}|^2 - 4\vec{p} \cdot \vec{q} - 3(4|\vec{p}|^2) = 0 \]
This simplifies to:
\[ 15|\vec{p}|^2 - 4\vec{p} \cdot \vec{q} - 12|\vec{p}|^2 = 0 \]
Thus:
\[ 3|\vec{p}|^2 = 4\vec{p} \cdot \vec{q} \implies \vec{p} \cdot \vec{q} = \frac{3}{4}|\vec{p}|^2 \]
(Mark this as Equation 2)

8. **Finding cosine of the angle**:
The dot product can also be expressed as:
\[ \vec{p} \cdot \vec{q} = |\vec{p}||\vec{q}|\cos(\theta) \]
Substituting from Equation 1:
\[ \frac{3}{4}|\vec{p}|^2 = |\vec{p}|(2|\vec{p}|)\cos(\theta) \]
Simplifying gives:
\[ \frac{3}{4}|\vec{p}|^2 = 2|\vec{p}|^2\cos(\theta) \implies \cos(\theta) = \frac{3}{8} \]

9. **Finding sine of the angle**:
Using the identity \( \sin^2(\theta) + \cos^2(\theta) = 1 \):
\[ \sin^2(\theta) = 1 - \left(\frac{3}{8}\right)^2 = 1 - \frac{9}{64} = \frac{55}{64} \]
Thus:
\[ \sin(\theta) = \sqrt{\frac{55}{64}} = \frac{\sqrt{55}}{8} \]

### Final Answer:
\[ \sin(\theta) = \frac{\sqrt{55}}{8} \]